description/proof of that for \(C^\infty\) vectors bundle, trivializing open subset is not necessarily chart open subset, but there is possibly smaller chart trivializing open subset at each point on trivializing open subset
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) vectors bundle of rank \(k\).
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
- The reader admits the proposition that for any map between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction or expansion on any codomain that contains the range is \(C^k\) at the point.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
- The reader admits the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) vectors bundle, a trivializing open subset is not necessarily a chart open subset, but there is a possibly smaller chart trivializing open subset at each point on any trivializing open subset.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\((E, M, \pi)\): \(\in \{\text{ the } C^\infty \text{ vectors bundles of rank } k\}\)
\(U\): \(\in \{\text{ the trivializing open subsets of } M\}\)
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Statements:
\(U\) is not necessarily any chart open subset
\(\land\)
\(\forall p \in U (\exists (U'_p \subseteq M, \phi'_p) \in \{\text{ the charts of } M \text{ around } p\} (U'_p \subseteq U \land U'_p \in \{\text{ the trivializing open subsets }\}))\)
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2: Proof
Whole Strategy: Step 1: cite an example \(U\) that is not any chart open subset; Step 2: around \(p \in U\), take a chart, \((U_p \subseteq M, \phi_p)\), and the chart, \((U'_p := U_p \cap U \subseteq M, \phi'_p := \phi_p \vert_{U'_p})\); Step 3: take a trivialization, \(\Phi: \pi^{-1} (U) \to U \times \mathbb{R}^k\), and take the trivialization, \(\Phi \vert_{\pi^{-1} (U'_p)}: \pi^{-1} (U'_p) \to U'_p \times \mathbb{R}^k\).
Step 1:
\(U\) is not necessarily a chart open subset, because \(U\)'s being a trivializing open set does not guarantee that \(U\) is a chart open set.
For example, for the product bundle, \(M \times \mathbb{R}^k\), \(U = M\) is a trivializing open subset, but \(M\) does not necessarily have a global chart, so, \(U\) is not necessarily a chart open set.
Step 2:
Around \(p \in U\), there is a chart, \((U_p \subseteq M, \phi_p)\).
\((U'_p := U_p \cap U \subseteq M, \phi'_p := \phi_p \vert_{U'_p})\) is a chart, by the proposition that for any \(C^\infty\) manifold with boundary and its any chart, the restriction of the chart on any open subset domain is a chart.
Step 3:
There is a trivialization, \(\Phi: \pi^{-1} (U) \to U \times \mathbb{R}^k\).
\(U'_p\) is a trivializing open subset with the trivialization, \(\Phi \vert_{\pi^{-1} (U'_p)}: \pi^{-1} (U'_p) \to U'_p \times \mathbb{R}^k\), by the proposition that any open subset of any \(C^\infty\) trivializing open subset is a \(C^\infty\) trivializing open subset.
3: Note
If \(U\) is not any chart open subset on \(M\), \(\pi^{-1} (U)\) is not necessarily a chart open subset on \(E\), while \(\pi^{-1} (U'_p)\) is a chart open subset on \(E\), by the proposition that for any \(C^\infty\) vectors bundle, the trivialization of any chart trivializing open subset induces the canonical chart map.
